Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation

TL;DR

This paper studies the convergence to equilibrium and particle approximation for a Vlasov-Fokker-Planck equation, providing explicit rates and probabilistic bounds for the system's behavior.

Contribution

It introduces a probabilistic approach to prove exponential convergence, propagation of chaos, and quantitative deviation bounds for the particle approximation of the equation.

Findings

Explicit exponential convergence rate to equilibrium.

Propagation of chaos with convergence rates.

Quantitative deviation bounds for particle approximation.

Abstract

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of the equilibrium solution of the equation by an empirical mean of the particles at given time.